The Book Review Column1
by William Gasarch
Department of Computer Science
University of Maryland at College Park
College Park, MD, 20742
email: gasarch@cs.umd.edu
Welcome to the Book Reviews Column. We hope to bring you at least two reviews of books every
month. In this column three books are reviewed.
1. Complexity Theory Retrospective II, edited by: Lane A. Hemaspaandra and Alan L.
Selman. Reviewed by: Eric Allender. This book is (mostly) a collection of surveys that whose
intention is to introduce the reader to several areas of complexity theory.
2. Basic Simple Type Theory by J. Roger Hindley. Reviewed by Brian Postow. This book
is an introduction to type theory.
3. Discrete Mathematics in the Schools, Edited by Joseph G. Rosenstein, Deborah S. Franzblau,
and Fred S. Roberts. Reviewed by Neal Koblitz. This is an unusual book for this column;
however, I feel that its contents will be of interest to this community. There is a debate going
on about teaching discrete math in the high schools. This book is from a workshop on the
topic. The workshop was clearly on the YES side of the debate. Neil Koblitz reviews the
book and also interjects his own opinions.
We are looking for reviewers for the following books: (1) Information flow: The logic of
distributed systems by Barwise and Seligman, (2) Control flow semantics by Bakker and Vink. I
can be contacted at the email address above. (3) Probabilistic Combinatorics and its applications
edited by Bela Bollobas. In exchange for a review you get a FREE COPY!
Review of: “Complexity Theory Retrospective II2 ,
Edited by: Lane A. Hemaspaandra and Alan L. Selman
Publisher: Springer Verlag
Reviewed by: Eric Allender, Rutgers University
1
Overview
Complexity theory is a rapidly changing and expanding field. Textbooks in complexity theory are
soon out-of-date – often before they are even published. Consequently, it can be difficult to know
where to look to get an overview of the field.
Thus the collection of well-written survey articles in this volume edited by Hemaspaandra and
Selman is especially welcome. This Retrospective II is the companion to an earlier Complexity Theory Retrospective published in 1990. That earlier Retrospective consisted mostly (but not entirely)
of expanded and polished versions of expository presentations from the series of IEEE Conferences on Structure in Complexity Theory (since re-named the IEEE Conference on Computational
1
2
c William Gasarch, 1998.
c Eric Allender, 1998
2
Complexity). This current volume also contains some papers that began life as Structures presentations, but contains a larger proportion of new papers, written expressly for Complexity Theory
Retrospective II.
Although this volume contains surveys on a very wide variety of topics, it is not intended to
be encyclopedic. Certainly there are many important and active areas of complexity theory that
are not discussed at all. In the paragraphs that follow, I briefly discuss each contribution in turn.
It should be apparent to the reader that many of the most important recent developments in the
field are covered by this collection.
2
Summary of Contents
Time, Hardware, and Uniformity. By David Mix Barrington and Neil Immerman. Descriptive complexity has enriched the studies of logic and of circuit complexity. In this survey, the
authors cover the latest developments in this field, showing how various parameters in the definition of classes of logical formulae correspond to fundamental resources in the definition of uniform
complexity classes.
Quantum Computation. By André Berthiaume. The title says it all. This is an excellent
introduction to the new and rapidly-changing area of quantum computation, starting from the
very beginning, and continuing through Shor’s factoring algorithm. A discussion of the physical
possibility of realizing a quantum computer is also included. The author informs me (personal
communication) that a follow-up survey on recent developments is already needed. (Two typos
worth mentioning: the summation on the bottom of page 33 should be
|ψi =
1
X
i,j=0
αi βj |iji
and “f (j)” should be replaced by “f (i)” in the expression in line 5 of page 42.)
Sparse Sets versus Complexity Classes. By Jin-Yi Cai and Mitsunori Ogihara. The study
of complete sets is a central topic in complexity theory. In particular, a question of long-standing
interest concerns the issue of whether a complete set can be sparse. Some lovely work in the 1990’s
led to rapid progress on this front (for various complexity classes and notions of reducibility). This
is a survey of this work by some of the people most involved.
Counting Complexity. By Lance Fortnow. “Counting” complexity classes are classes defined
in terms of the number of satisfying assignments of nondeterministic Turing machines. Although
counting classes have been around almost since the beginning of complexity theory, our understanding of the properties of these classes has grown dramatically in the past decade. Fortnow surveys
these developments, in some cases providing new and simplified proofs.
A Taxonomy of Proof Systems. By Oded Goldreich. It is no secret that the study of interactive
proof systems is one of the big success stories in the recent history of complexity theory. There has
been so much happening that it can be hard to keep it all sorted and in perspective. This survey
by Goldreich is a very welcome contribution.
Structural Properties of Complete Problems for Exponential Time. By Steven Homer.
It is important to understand the class of NP-complete sets. In order to get a hint of what their
structure might be like, it has been instructive to consider the complete sets for (deterministic
3
and nondeterministic) exponential time. Homer explains how – in contrast to the polynomialtime setting – current techniques have been sufficient to give a fairly clear picture of the shape of
complete sets in the exponential-time world.
The Complexity of Obtaining Solutions for Problems in NP and NL. By Birgit Jenner
and Jacobo Torán. Most effort in complexity theory has concentrated on the complexity of zero-one
valued functions (decision problems). The complexity of other functions (such as the complexity
of finding optimal solutions for NP search problems) has received less attention, but there is a
significant and growing body of work that does deal explicitly with this topic. Jenner and Torán
discuss the issues involved in capturing the relevant notions of complexity, and survey the most
important results in this area.
Biological Computing. By Stuart A. Kurtz, Stephen R. Mahaney, James S. Royer, and Janos
Simon. In contrast to the other papers in this volume, the article on Biological Computing is
not a survey of recent work in the area. Instead, after presenting a brief introduction to some
relevant aspects of biochemistry, the authors present a speculative new approach to using biological
machinery to build single-molecule processors, as an alternative to Adleman’s model. This makes
for provocative and interesting reading.
Computing with Sublogarithmic Space. By Maciej Liśkiewicz and Rüdiger Reischuk. Complexity theory has been unable to resolve the basic open questions about the relative power of time,
space, nondeterminism, and alternation. There are two notable exceptions, however:
• For constant-depth, polynomial-size circuits (or equivalently, log-time alternating Turing machines making a constant number of alternations), there is an infinite hierarchy. More alternations yield more power.
• When time is unrestricted, but less than logarithmic space is available, there is also a hierarchy. More alternations yield more power.
Liśkiewicz and Reischuk have been very active in discovering the true power of alternation in this
second setting (the sublogarithmic space world), and this very well-done survey presents the results,
the intuition behind the proofs, and most of the proof techniques.
The Quantitative Structure of Exponential Time. By Jack H. Lutz. Resource-bounded
measure theory has been around for over a decade, and the number of interesting results in this
area has been growing until now more and more people are realizing that they should find out
what’s going on with measure in complexity theory. This fine survey by Lutz (the main force
behind resource-bounded measure) is a good introduction to the area and a useful resource.
Polynomials and Combinatorial Definitions of Languages. By Kenneth W. Regan. Algebraic tools were behind a number of the most important advances in complexity theory during
the past decade. In particular, it turned out to be extremely useful to consider various ways of
representing functions by polynomials in various ways. Regan gathers all of this diverse material
together and presents it in a format and organization that will be very useful to students and others
wanting to gain a mastery of these techniques.
Average-Case Computational Complexity Theory. By Jie Wang. In spite of the good news
concerning the growing number of heuristics for solving NP-complete problems that work well in
practice, there is evidence that many NP-complete problems really are intractable in any practical
sense. The theory of average-case computational complexity is vitally important, as the branch
of complexity theory that seeks to address practical considerations of real-world performance (as
4
opposed to worst-case guarantees) in a rigorous way in an effort to prove lower bounds. Surveys
have played a big role in the development of this field, and Wang’s survey in this volume does an
excellent job of motivating the definitions and covering the most important results in the area.
3
Opinion
This is a really fine collection of papers. All of the articles are of uniformly high quality, and all of
the topics are important.
I would recommend this volume as supplementary course material for a graduate course or
seminar on complexity theory, and I would also recommend it to every active researcher in the
field. In most cases, people really wanting to completely understand the proofs of the main theorems
presented will have to consult the original sources as well, but these surveys do an excellent job of
telling the reader what those important theorems are, and where to go to find the best proofs.
Review of
Basic Simple Type Theory3
Series: Cambridge Tracts in Theoretical Computer Science #42
Author: J. Roger Hindley
Publisher: Cambridge University Press, 1997
Reviewer: Brian Postow
4
Overview
Type theory is a field that has implications in several areas of mathematics and computer science. It
was first invented in the context of set theory as a method of getting around several paradoxes that
were discovered during the beginning of this century. Its relationship to the theory of computation
through the lambda calculus was quickly realized, followed soon after by links to logic and proof
theory. More recently, the effects of type theory have been felt in many different fields of computer
science, from increased understanding of complex type systems used in object-oriented languages
to the type deduction used in ML and other modern functional languages. This book is a gentle
introduction to the area for those who know a little λ calculus. It attempts to give a flavor of the
theory and explain some of the more interesting results, including the type deduction algorithm
that was partially developed by the author and is now used in ML.
• Chapters 1 and 2 present the basic underlying definitions and background of type theory. They
give a brief introduction to the λ calculus, and add types to λ terms.
• Chapters 3, 7 and 8 present the major algorithms in the text: assigning a type to a λ term,
finding a term of a given inhabited type, and counting the number of inhabitants of a given
type.
• Chapters 4 and 5 add some functionality and new syntax to the type system that is being
developed.
• Chapter 6 is a digression about the relationship between type theory and logic.
• Chapter 9 contains proofs of several details that are glossed over in the rest of the book.
3
c Brian Postow, 1998
5
5
Summary of Contents
Chapter 1 very briefly describes the untyped λ calculus. It provides all of the standard definitions
(terms, abstractions, α, β and η reductions, etc) a few versions of the Church-Rosser theorem, and
three restrictions of the λ calculus that are examined later. Since the purpose of the book is to
give a flavor of the systems used, and as the first chapter just provides background, proofs in this
chapter are sketchier than in the rest of the book. Several proofs are omitted entirely, replaced
instead by references to papers containing the complete proofs.
Chapter 2 adds types to the λ calculus. It describes both the Church approach of typed terms,
and the Curry approach of type-assignments and deductions. The majority of the book deals with
the Curry model; however, a brief foray into typed-terms is made in chapter 5.
The main purpose of the chapter is to define the system T Aλ which is the type and deduction
system that is used in the rest of the book, albeit sometimes with added restrictions. T A λ is a
typed calculus with axioms:
x : τ 7→ x : τ
for each term-variable x and each type τ . and two deduction rules:
(→ E)
Γ1 7→ P : (σ → τ )
Γ2 7→ Q : σ
Γ1 ∪ Γ2 7→ (P Q) : τ
and
(→ I)
Γ 7→ P : τ
Γ − x 7→ (λx.P ) : (σ → τ )
Where Γ is a type context, an association of terms to types. In (→ E), Γ1 and Γ2 can’t disagree
on what type gets mapped to any terms, and in (→ I), Γ must be consistent with x : σ.
From these two rules and the axioms, we can define a set of terms that are typable in T A λ .
Many untyped terms, such as the Y combinator: (λx.xx)(λx.xx), can not be given a type, so the
set of typable terms is non-trivial.
The rest of the chapter describes how the reductions change with the addition of types, and
proves various normalization theorems.
Chapter 3 defines the principal type of a term, the most general type that is assignable to a
given term. Then, it proves that every typable term has a principal type by providing an algorithm
that given an untyped term produces its principal type, or, if the term is untypable, says so. The
majority of the chapter is taken up by describing the algorithm and proving its correctness.
In Chapter 2 it was noted that, even though two terms may be β equivalent, it is possible that
they have no types in common. Chapter 4 explores the system T Aλ+βη which ensures that if two
terms are β equivalent, they have the same types by adding the following rule:
(Eqβ )
Γ 7→ M : τ
M =β N
Γ 7→ N : τ
This rule increases the number of typable terms, for example, making:
(λuv.v)((λx.xx)(λx.xx))
typable, since it β reduces to λv.v which has type τ → τ for any τ . Obviously this is not
typable in the previous system of T Aλ , because the second term is the Y combinator which is
not typable. The rest of the chapter is devoted to describing other changes that this added rule
creates, including the presence of weak normalization, but the loss of strong normalization.
6
Chapter 5 is a brief foray into typed-terms. The goal of the chapter is not to compare the Curry
and Church approaches. Instead typed terms are used only as an alternative notation to describe
the type deductions used in the Curry approach. The chapter mainly redefines the terms given in
chapter 2, but using typed terms instead of type assignments. It also briefly describes how a typed
term will be used to describe a deduction.
Chapter 6 is the obligatory digression into intuitionistic implicational logic. It describes the
logic, a little bit of its history, and its proof method. It also discusses and proves the Curry-Howard
Theorem, that the provable formulae of intuitionistic implicational logic are exactly the types of
closed λ terms, and that there is a one-to-one correspondence between T A λ type deductions and
natural deductions in the intuitionist implicational logic.
In addition, three weaker logics are introduced that have the same relationship with the weaker
versions of T Aλ introduced in chapter 1.
Finally a version of Hilbert style logic (classical sentential logic) is described that can be expressed in terms of intuitionistic logic, and equivalent results are proved for it.
Chapter 7 proves the converse of chapter 3, that given a type τ , and a term of that type, M ,
there is a term for which τ is the principal type. To prove this the book gives an algorithm that
given τ and M , finds such a term, and proves the algorithm’s correctness. Equivalent theorems are
proven for the weaker theories. Chapter 6 is referenced here because the proof goes back and forth
between the λ calculus and intuitionistic logic.
Chapter 8 answers the question “Given a type τ how many terms can receive τ in T A λ ?”
Obviously if τ is inhabited (ie, there is at least one term that can be assigned type τ ) there will be
an infinite number of terms because we can always apply the identity function to the term without
changing the type. However, the question of how many terms in β-normal form can receive a given
type is more interesting. The answer to this question can be infinite, any finite number, or even 0
(even if the type is inhabited, it may have no β-normal terms). Most of the chapter is devoted to
describing and proving the correctness of an algorithm that, given a type, outputs the number of
β normal terms of that type and enumerates them.
In an attempt to keep the majority of the proofs short and readable the author has chosen to
eliminate some details from the actual proofs. However, in order to maintain rigor he includes all
of these details in chapter 9. The final chapter contains details about the exact structure of terms,
types and deductions, and various other details that were glossed over in other parts of the book.
Each section carefully describes which chapter it was meant to be read with.
6
Style
This book is relatively easy to read. The proofs are for the most part short, with references to more
complete proofs in the literature. Every section is clearly labeled so you never loose track of where
you are or where you are going. In particular, the author labels many subsections with Note for
important relationships that you might have missed, and (more interestingly) Warning for results
that you might assume follow from a given theorem but don’t.
7
Opinion
This is an excellent introduction to type theory. It doesn’t bog the reader down in any of the
messy details of the proofs (unless he reads chapter 9) and yet it provides many of the most
interesting results in the field. It has some exercises, a few of which have solutions in the back, and
7
a comprehensive bibliography. Overall it is a great book for someone who wants to get his feet wet
in type theory, but doesn’t want to get in over his head.
Review of
Discrete Mathematics in the Schools 4
DIMACS Series in Discrete Mathematics and
Theoretical Computer Science, Volume 36, 1997
Editors: Joseph G. Rosenstein, Deborah S. Franzblau, and Fred S. Roberts
Publisher: American Mathematical Society
Reviewer: Neal Koblitz
8
Overview
In response to the recommendations of the National Council of Teachers of Mathematics [9] and the
increasing interest on the part of educators and scientists in the teaching of discrete mathematics
in the schools, the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)
of Rutgers University held a conference on the subject on October 2–4, 1992. The volume under
review grew out of that conference. The five-year delay is explained by the editors’ decision to
solicit expanded articles and additional contributions from people who have played leadership roles
in introducing discrete math at precollege levels. The book is well worth the wait. The 34 articles
are well written, carefully edited, and full of valuable ideas.
The collection is divided into eight sections as follows:
1.
2.
3.
4.
5.
6.
7.
8.
The Value of Discrete Mathematics: Views from the Classroom
The Value of Discrete Mathematics: Achieving Broader Goals
What Is Discrete Mathematics: Two Perspectives
Integrating Discrete Mathematics into Existing Mathematics Curricula, Grades K–8
Integrating Discrete Mathematics into Existing Mathematics Curricula, Grades 9–12
High School Courses on Discrete Mathematics
Discrete Mathematics and Computer Science
Resources for Teachers (discussions of books, videos, software, and the Leadership Program in
Discrete Mathematics of Rutgers University)
I cannot do justice to all 34 articles in the space available. In the next section I will describe
what I found especially interesting in a few of the articles.
9
Partial Summary of Contents
Joseph Rosenstein’s “Introduction” includes a succinct explanation of how discrete math can revitalize school mathematics:
Discrete mathematics is:
Applicable: In recent years, topics in discrete mathematics have become valuable tools
and provide powerful models in a number of different areas.
4
c Neil Koblitz, 1998
8
Accessible: In order to understand many of these applications, arithmetic is often
sufficient, and many others are accessible with only elementary algebra.
Attractive: Though easily stated, many problems are challenging, can interest and
attract students, and lend themselves to exploration and discovery.
Appropriate: Both for students who are accustomed to success and are already contemplating scientific careers, and for students who are accustomed to failure and perhaps
need a fresh start in mathematics. (pp. xxvi–xxvii)
In Section 1, Susan Picker describes her experiences with students of the latter sort. In a class
of remedial tenth-grade students in Manhattan, she reports remarkable success working with graph
coloring to model scheduling conflicts. Not only did her students develop some real “expertise” in
this type of problem; more importantly, their whole perception of mathematics and of themselves
as students improved dramatically.
In Section 2, Nancy Casey and Michael Fellows use examples from graph theory and knot theory
to explain how to introduce exciting mathematical notions in the early grades. They argue that
children in K–4 should experience:
• a surprising mathematical truth that contradicts intuition;
• a simply-stated mathematical problem with no known solution;
• logical paradox;
• the notion of a limit;
• mathematical exploration.
They illustrate how elementary activities can give kids a foretaste of such fundamental topics as
mathematical proof, algorithmic efficiency, unsolved problems, and one-way functions.
The authors argue that mathematics should be seen as a kind of “literature” whose value goes
far beyond its everyday utility, and they say that children should be exposed to the frontiers of
knowledge. In another article [1], Fellows gives examples of how work with young children can
even be a stimulus to one’s own research in discrete mathematics. (See also [2] and [7] for similar
discussions in the context of cryptography.)
Two other articles in Section 2 that I particularly enjoyed were Henry Pollak’s piece on mathematical modeling and Fred Roberts’ discussion of “The Role of Applications in Teaching Discrete
Mathematics.” Pollak gives a clear explanation, with examples, of what is really taking place
when we use mathematics to try to understand real-world phenomena. Roberts gives nine “rules
of thumb” to guide teachers in their classroom use of practical applications, and he illustrates
these rules through a number of vivid examples based on the Traveling Salesperson Problem, graph
coloring, and Euler paths.
Section 3 contains the longest article in the book — the full text of the chapter on discrete
mathematics in the New Jersey Mathematics Curriculum Framework, divided into sections for
each of the K–2, 3–4, 5–6, 7–8, and 9–12 grade levels. The author is Director of the New Jersey
Mathematics Coalition, as well as one of the editors of the book under review.
One of the strengths of the book is the inclusion of reports by in-service teachers. For example,
in Section 5, Bret Hoyer (of Cedar Rapids, Iowa) relates “A Discrete Mathematics Experience
with General Mathematics Students”; and in Section 6, Charles Biehl (of Wilmington, Delaware)
describes his high school “math analysis” course designed for students who are not likely to become
math or science majors at college.
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10
Opinion
A compelling case can be made for the use of discrete math in certain situations:
1. Classroom visits by scientific researchers. Too often, scientists-in-the-schools programs amount
to little more than “show and tell,” with the students learning no basic science. If, on the other
hand, the scientists introduce discrete math activities to the children, then the kids will develop
a deeper appreciation of scientific modes of thinking.
2. Special programs for talented students, run by highly motivated teachers.
3. Programs for slow students who have given up on the standard math subjects. See especially Susan Picker’s article “Using Discrete Mathematics to Give Remedial Students a Second Chance”
and Charles Biehl’s article “A Fresh Start for Secondary Students.”
It is not so clear that a new emphasis on discrete math should be mandated for all students
throughout the school system. The authors of Discrete Mathematics in the Schools are dedicated
and talented pedagogues. What works well for them might not turn out so well in the hands of an
average teacher.
A common explanation for the disappointing performance of American youngsters in many
international comparisons (most recently, in the Third International Math and Science Study) is
that the mathematics curriculum in U.S. schools is “a mile wide and an inch deep.” That is, every
year students glide rapidly through a large number of topics without developing a mastery of any
of them.
Most of the authors of the collection under review seem to me to be insufficiently aware of the
dangers of handing teachers a new list of concepts and activities to shoe-horn into the curriculum.
Given the problems and pressures that confront the average teacher in America, it is not likely that
the introduction of discrete math on a massive scale would go as well as the authors imagine.
The main danger I see is that, if handled poorly, a push for discrete math could contribute
to a further “dumbing down” of the math curriculum. Responding to pressure from parents and
politicians, teachers and textbooks will coddle the children with easy material that is not ageappropriate. If, for instance, an 8th-grade textbook consists mostly of material that should be easy
for a 5th grader and contains only a small amount of material that one would think of as challenging
at the 8th-grade level, then a teacher who is not careful can easily spend the whole year doing only
the part of the book that would make an appropriate 5th-grade textbook.
Students will get high test scores and inflated grades, will never develop self-discipline and
good study habits, and will enter college unprepared for university-level mathematics and science
courses. This would, of course, be the exact opposite of what the authors of Discrete Mathematics
in the Schools intend. But as I read the article by Joseph Rosenstein taken from the New Jersey
Mathematics Curriculum Framework — offering a grab bag of nice topics and activities for students
to work on — I was struck by the absence of any concrete indication of what constitutes reasonable
measures of satisfactory performance. I could not help noting how easy it would be for teachers and
textbook writers to use toned-down versions of the activities that require little sustained mental
effort on the part of the youngsters.
I know of only one systematic attempt to address the question of assessment of student achievement in discrete math: the book Measuring Up by the Mathematical Sciences Education Board
[8]. Like the volume under review, Measuring Up has some excellent material; in fact, I use it as
a required text in a course I teach for undergraduate math education majors. But unfortunately,
most of the assessment standards are, in my judgment, too simple-minded for the intended age
level.
10
When I travel to different parts of the world, I like to develop contacts in the local schools. I
often visit math classes to give a workshop in discrete math to children between 9 and 13 years old;
I have done this in about a dozen countries of Asia, Africa, and the Americas. Some of the activities
I enjoy sharing with the youngsters are discussed either in the book under review or in [8]. But I
always use versions that are more sophisticated and challenging than the “assessment prototypes”
in [8], because foreign children would find the American versions too easy and simplistic — not
challenging enough to be interesting.
Let me give an example. One of the activities in [8] is an arithmetic dice game, played as follows.
The children roll 3 dice, find expressions for the integers 1 through 10 in terms of the numbers on
the dice and the operations +, −, × and ÷ (using each number exactly once), and see how many
of the numbers
7
8
6
9
5
3
10
4
2
1
can be “knocked down” (by analogy with bowling). For more advanced students, the suggestion in
[8] is to use 4 dice and put another row of pins numbered 11 through 15 at the top.
Here is the modification that I use when I work with 4th to 7th graders in other parts of the
world: Roll 5 dice, allow +, −, ×, ÷ and squaring (and repeated squaring) of any number or
expression, and have the kids generate prime numbers, each one larger than the ones before. Once
they get numbers above 1000, testing for primality becomes harder. At the end I ask for a show
of hands: How many think that the game could go on forever (generating an infinite sequence of
primes)? How many don’t? This is an unsolved problem of number theory (a generalization of the
Fermat prime problem). This prime number dice game has worked well both with average kids (in
Belize, Central America, and in Cape Town, South Africa, for example) and with unusually bright
kids (in a school in Vietnam). This version of dice arithmetic is more interesting and challenging
than the mickey-mouse version in [8].
11
Conclusion
The book Discrete Mathematics in the Schools is full of useful material and thought-provoking
discussion. Even though it does not answer all questions one might have about the use of discrete
math on a massive scale in the schools, it is a valuable first step. All scientists who are interested
in improving K–12 math education should be sure to read this book.
References
[1] M. R. Fellows. Computer science and mathematics in the elementary schools, in N. D. Fisher,
H. B. Keynes, and P. D. Wagreich, editors, Mathematicians and Education Reform 1990–1991.
Providence RI: Amer. Math. Society, 1993, 143–163.
[2] M. R. Fellows and N. Koblitz. Kid Krypto, in E. F. Brickell, editor, Advances in Cryptology –
Crypto ’92. New York: Springer-Verlag, 1993, 371–389.
11
[3] M. R. Fellows and N. Koblitz. Math Enrichment Topics for Middle School Teachers, 1995.
[4] S. Garfunkel et al. For All Practical Purposes: Introduction to Contemporary Mathematics,
3rd edition, New York: W. H. Freeman and Company, 1994.
[5] Allyn Jackson. The Math Wars: California Battles It Out over Mathematics Education Reform,
Notices of the Amer. Math. Society 44, No. 6 & 7, 1997, 695–702 & 817–823.
[6] N. Koblitz. The Case Against Computers in K–13 Math Education (Kindergarten through
Calculus), The Mathematical Intelligencer 18, No. 1, 1996, 9–16.
[7] N. Koblitz. Cryptology As a Teaching Tool, Cryptologia 21, 1997, 317–326.
[8] Mathematical Sciences Education Board and National Research Council. Measuring Up: Prototypes for Mathematics Assessment, Washington: National Academy Press, 1993.
[9] National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School
Mathematics, Reston VA: NCTM, 1989.
12